Custom multivariate discrete distribution [closed]

This is related to How to define an n-variate empirical distribution function probability for any n?
and
RandomVariate from 2-dimensional probability distribution
but I don’t think neither questions (and their answers) answer the question below. (The first one constructs the problem from data and not from a priori specified probability weights, and hence they can use NProbability; and the second one seems like an extreme overkill for something like a simple discrete distribution where one doesn’t really need to kick out random number generators).

I want to construct a multivariate discrete distribution so that I can use the full functionality of RandomVariate and things of that sort.

In 1-dimension, I can use EmpiricalDistribution. For instance, for a X∼Bernoulli(p)X∼Bernoulli(p)X \sim Bernoulli(p) with p=1/2p=1/2p = 1/2, it is simply

gdist = EmpiricalDistribution[{0.5, 0.5} -> {0, 1}]

and from this, I can go on to compute mean and variances via Expectation, say

Expectation[ 2*x + 1, x \[Distributed] gdist]

as afforded by RandomVariate and all its friends.

Question: How does one do that for a multivariate discrete distribution (whether via EmpiricalDistribution or not)? That is, suppose we consider,

where of course p10+p01+p00+p11=1p_{10} + p_{01} + p_{00} + p_{11} = 1 are the probability weights. How does one implement the above distribution, say labelled as gmultdist so that we can compute Expectation[ 2*x + 3*y, {x,y} \[Distributed] gmultdist]?